Arithmetic Sequence Calculator
Calculate n-th term, sum, and generate sequence terms for arithmetic progressions
Starting value of the sequence
Difference between consecutive terms
Which term to find
How many terms to include
Quick Examples
Term # (a) Sum of Terms (S) Arithmetic Sequence
Step-by-Step Calculation
n-th Term
a
Sum of Terms
S
Common Difference
d
Common Arithmetic Sequences
| Name | First Term (a₁) | Difference (d) | Sequence |
|---|---|---|---|
| Natural Numbers | 1 | 1 | 1, 2, 3, 4, 5... |
| Even Numbers | 2 | 2 | 2, 4, 6, 8, 10... |
| Odd Numbers | 1 | 2 | 1, 3, 5, 7, 9... |
| Multiples of 5 | 5 | 5 | 5, 10, 15, 20, 25... |
| Countdown from 100 | 100 | -10 | 100, 90, 80, 70... |
| Negative Sequence | -5 | 3 | -5, -2, 1, 4, 7... |
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About Arithmetic Sequence Calculator
What is an Arithmetic Sequence?
An arithmetic sequence (also called an arithmetic progression) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference (d).
Example: 2, 5, 8, 11, 14... is an arithmetic sequence with first term a₁ = 2 and common difference d = 3.
How to Use This Calculator
- Select a calculation mode - Choose whether to find the n-th term, calculate the sum, or generate a sequence
- Enter the first term (a₁) - The starting value of your sequence
- Enter the common difference (d) - The constant value added to each term
- Enter n - The term position or number of terms
- View results - See the calculated value with step-by-step explanation
Key Formulas
n-th Term Formula
aₙ = a₁ + (n - 1) × d
Where:
- aₙ = the n-th term
- a₁ = the first term
- n = the term position
- d = the common difference
Sum of n Terms Formula
Sₙ = n/2 × (a₁ + aₙ) or Sₙ = n/2 × (2a₁ + (n - 1) × d)
Where:
- Sₙ = sum of the first n terms
- n = number of terms
- a₁ = first term
- aₙ = n-th term
- d = common difference
Common Applications
| Application | Example |
|---|---|
| Salary increases | Annual fixed raise amounts |
| Seating arrangements | Rows with fixed additional seats |
| Stair steps | Each step adds same height |
| Loan payments | Fixed principal reduction |
| Distance intervals | Equal spacing between points |
Examples
Finding the 10th Term
For sequence starting at 3 with d = 4: a₁₀ = 3 + (10-1) × 4 = 3 + 36 = 39
Finding the Sum of First 8 Terms
For sequence 2, 5, 8, 11...: S₈ = 8/2 × (2×2 + 7×3) = 4 × (4 + 21) = 100
Frequently Asked Questions
What's the difference between arithmetic and geometric sequences?
An arithmetic sequence has a constant difference between terms (add the same value). A geometric sequence has a constant ratio between terms (multiply by the same value).
Can the common difference be negative?
Yes! A negative common difference creates a decreasing sequence. For example: 20, 17, 14, 11... has d = -3.
How do I find the common difference if I know two terms?
Subtract the earlier term from the later term, then divide by the number of steps between them: d = (aₘ - aₙ) / (m - n).
What if the first term is 0?
That's perfectly valid! For example, 0, 5, 10, 15... is an arithmetic sequence with a₁ = 0 and d = 5.
Tip: Arithmetic sequences are everywhere in real life—from numbered parking spots to saving a fixed amount each month!